The conformal radius as a function and its gradient image

Abstract

Let Ω be a domain in ℂ̄ with three or more boundary points in ℂ̄ and R(w, Ω) the conformal, resp. hyperbolic radius of Ω at the point w ∈ Ω \{∞}. We give a unified proof and some generalizations of a number of known theorems that are concerned with the geometry of the surface SΩ = {(w,h) | w ∈ Ω, h = R(w, Ω)} in the case that the Jacobian of ▽R(w, Ω), the gradient of R, is nonnegative on Ω. We discuss the function ▽R(w, Ω) in some detail, since it plays a central role in our considerations. In particular, we prove that ▽R(w, Ω) is a diffeomorphism of Ω for four different types of domains.